Page 32 - Fall 2006

Fig. 18. Circumferential harmonics of the surface-averaged accelerance of a short 3 mm thick, 1.2 m long, 0.8 m diameter steel cylinder.
Fig. 16. Example of an n=4 circumferential mode in a cylindrical shell.
sine waves around the circumference.
When bending waves travel around a structure with cur-
vature, they are coupled strongly to in-plane longitudinal waves, which travel at much higher speeds than the bending waves do (recall our discussions of wave speeds earlier). The coupling occurs because curvature causes the transverse dis- placement to be resisted not only by flexural stiffness, but also by the membrane, or ‘hoop’ stiffness. Bending waves sped up by membrane stiffness resonate at higher frequencies than they would in an equivalent flat plate.
The theory representing coupled flexural/membrane waves is far too complicated to include here, and I refer you to the compendium by Leissa24 for the various approaches that have been developed to date. We can, however, look at some of the implications of the phenomena we have dis- cussed so far.
The circumferential harmonics of most interest to struc- tural acousticians are low order, and shown in Fig. 17. Recall that the circumferential motion of a cylindrical shell is com- prised of cos(nθ) and sin(nθ) components, where n is the num- ber of waves around the circumference for a given harmonic. The harmonic that radiates the most sound is n=0, where there is no sinusoidal variation of the wave around the circumfer- ence. The n=0 modes of a cylindrical shell are clustered around the ring frequency, which is the frequency at which a mem- brane wave is continuous around the circumference:
(26)
The ring frequency is sometimes expressed in dimen- sionless form, normalized to the radius a and longitudinal wave speed: Ωr = 2πfra/cl. n=0 modes are sometimes called ‘breathing’ modes, and as you might imagine, radiate sound
Fig. 17. Circumferential deformation of low-order modal harmonics: breathing, or n=0 circumferential motion (left); beam, or n=1 circumferential motion (center); and ovaling, or n=2 circumferential motion (right).
very efficiently.
n=1 motion is a single wave around the circumference,
and it appears that the cylinder cross section vibrates as a rigid body about the undeformed shape. n=1 motion gener- ally occurs in long cylinders, and can be represented more simply using beam theory, with the area and inertia of the cylinder cross-section input to the beam equations. For long cylinders (usually called pipes), beam theory can be used to model vibrations up to the frequency where the higher order harmonics cut on. In short cylinders, though, the beamlike n=1 modes do not appear until high frequencies.
When harmonics above n=1 appear in a cylinder’s vibra- tion, they are called ‘lobar’ modes. The first lobar mode is n=2, where the cross section deforms as an oval. Ovaling is often the first mode to cut on in cylinders of medium length.
An example of the surface averaged mobility of a medi- um length cylindrical shell is shown in Fig. 18. As with the flat plate mobility, the contributions to the mobility by the first several shell modes are shown in the figure, except this time, the mode orders do not increase with increasing fre- quency. In fact, the first shell mode to cut on is the n=5 mode, followed by the n=4, 6, and 3 harmonics. This behavior is typical of cylindrical shells, where lower order harmonics often cut on at frequencies above those of higher order har- monics. In this example, the breathing and beam modes have not yet cut on for frequencies up to 500 Hz (they do cut on eventually, though).
Since for pipes the n=1 beamlike modes dominate low frequency response, pipe mobilities can be approximated with infinite beam mobilities until the lobar modes cut on. A common approximation for the lobar mode cut on frequen- cy is Ω>0.77 h/a, where h is the thickness (recall Ω is 2πfa/cl). Another infinite mobility equation is used for shells above the lobar mode cut on frequency; and for dimensionless fre- quencies above 0.6, the cylindrical shell mobility approaches that of a flat plate without curvature. This is because as the flexural wavelengths become short with respect to the radius of curvature of the shell, the shell effectively becomes a flat plate and the flexural motion is uncoupled from the mem- brane motion.
So, the mobility Y of an infinite pipe (and the mean mobility of a finite one) may be computed as:
30 Acoustics Today, October 2006